Helical Networks of π‐Conjugated Rods – A Robust Design Concept for Bicontinuous Cubic Liquid Crystalline Phases with Achiral Ia 3¯d and Chiral I23 Lattice

Bicontinuous cubic liquid crystalline phases of π‐conjugated molecules, representing self‐assembled 3D‐ordered interpenetrating networks with cubic symmetry, are receiving increasing attention due to their capacity for charge transport in all three dimensions and their inherent spontaneous helicity. Herein, a robust general design concept for creating bicontinuous cubic phases is reported. It is based on a nonsymmetric‐substituted π‐conjugated 5,5′‐diphenyl‐2,2′‐bithiophene platform with one end containing three out‐fanning flexible chains and with a range of substituents at the other end (the apex). The cubic phases are stable over broad temperature ranges, often down to ambient temperature, and tolerate a wide range of apex substitution patterns, allowing structural diversity and tailoring of the cubic phase type and application‐relevant properties. With an increasing number and size of apex substituents, a sequence of three different modes of cubic self‐assembly is observed, following an increasing helical twist. Thus, two ranges of the achiral double network Ia 3¯d phase range can be distinguished, a long pitch and a short pitch, with the chiral triple network I23 cubic phase in the intermediate pitch range. The findings provide a new prospect for the directed design of cubic phase‐forming functional materials based on spontaneously formed helical network liquid crystals with tunable application specific properties.


Introduction
Networks are indispensable for information technology, intelligence, and open-end development of complex systems, [1,2] and covalently connected molecular networks form the basis of reticular chemistry. [3] Bicontinuous cubic phases (Cub bi ) Figure 1. The Cub bi phases formed by rod-like compounds; a) the double gyroid (Ia3d) involving two networks and b) the I23 phase formed by three networks; [87,99] c-f) representative examples for the major classes of rod-like compounds forming these two types of Cub bi phases; g,h) chiral conglomerate of the I23 phase of 3/ 35 I 2 as observed upon cooling at T = 125 °C between slightly uncrossed polarizers (contrast enhanced); the width of both images is 0.5 mm; i) the helical twist developing along the networks due to the clashing of the bulky end groups between the molecules organized in the networks shown in (a, b).
providing potential application relevant functional properties. [126][127][128] They are terminated at one end by a 3,4,5-tridecyloxybenzoyloxy group and at the opposite end, referred to as apex, by a series of systematically varied substituents to adjust their capability to self-assemble in network structures with cubic symmetry. In the series of compounds 3/ m X n , the apex is a benzoate unit and the position of the substituent(s) at this unit is indicated by superscripts (m) before X and the number of substituents is added as a subscript (n) after X, e.g., 3/ 35 I 2 means two iodines in positions 3 and 5. In compounds 3a/Y, the benzene ring at the apex is replaced by other aromatic, heteroaromatic, alicyclic, or branched units Y.

Synthesis
The synthesis of the tricatenar phenol 3b/OH (Scheme 1) was conducted by Suzuki type boronate cross-coupling reactions, as reported previously. [86] Acylation of 3b/OH with appropriate benzoyl chlorides led to the compounds 3/ m X n and 3a/Y, as described in the Supporting Information.

Investigation Methods
Investigation of the obtained materials was conducted by polarizing optical microscopy (POM) (Optiphot 2, Nikon microscope with a Mettler FP82HT heating stage), differential scanning calorimetry (DSC-7 and DSC-8000 Perkin Elmer, 10 K min −1 peak temperatures quoted in Table S1 in the Supporting Information), and X-ray diffraction (XRD). In-house XRD was carried out using Cu Kα radiation and a Vantec 500 area detector. High-resolution small-angle X-ray scattering patterns were recorded at beamline I22 of Diamond Light Source with a Pilatus 2M detector and at XmaS beamline B28 of the European Synchrotron Radiation Facility (ESRF). Powder samples in capillaries were temperature controlled with a Linkam hot stage.

Characterization of the Cubic Phases
The cubic phases appear uniformly dark between fully (90°) crossed polarizers as they are optically isotropic. The transition from the isotropic liquid (Iso) to the cubic phases was identified by the sudden increase of viscosity associated with a DSC peak with ΔH = 2-5 kJ mol −1 . Usually, there is a 5-12 K supercooling of the Iso-Cub transition relative to the Cub-Iso transition temperature on heating (Tables S1 and S2 in the Supporting Information for numerical data and Figures S1 and S14a in the Supporting Information for representative DSC traces). Polarizing microscopy allows an easy preliminary discrimination between the two cubic phase types formed by rod-like molecules, the Ia3 d and I23 phases. The achiral Ia3 d phase remains uniformly dark also after rotating one of the polarizers by a small angle out of the fully crossed orientation. By contrast, the mirror symmetry broken I23 phase shows a conglomerate of dark and bright domains after slight rotation of one polarizer by 1°-10° either clockwise or counterclockwise (Figure 1g,h).
For further confirmation, the cubic phases were investigated by XRD. The powder diffraction pattern of the Ia3 d phase is characterized by the well resolved (211) and (220) reflections, whereas for I23 a characteristic pattern of the dominating (321) with a shoulder for the (400) reflection and a small (420) reflection is observed ( Figure S9 and Tables S3 and S4, Supporting  Information). Representative examples were investigated using a synchrotron source (Figures S10-S12 and Tables S5-S9, Supporting Information). The high resolution XRD pattern of the cubic Ia3 d and I23 phases of compound 3/ 3 F and 3/ 35 F 2 , respectively, are shown in Figure 2a,c. The electron density (ED) maps of these two cubic phases, reconstructed on the basis of the diffraction intensities (Figure 2b,d) show the double gyroid structure of the Ia3 d phase and the complex triple network structure of the I23 phase. Details of the reconstruction are described in the Supporting Information, and for details of the ED reconstruction in the case of the non-centrosymmetric I23 phase, see also ref. [99]. Figure 3a shows the effect of increasing the size of apex substitution in the terminal 4-position with respect to the COO group (compounds 3/ 4 X). As shown, even for large substituents, like iodine and OCF 3 , the Cub bi phase is retained over a wide temperature range. It should be noted that the transition temperatures in Figure 3a and all following bar diagrams were measured on heating and the observed LC phases represent thermodynamically stable (enantiotropic) phases, whereas on cooling, much broader metastable cubic ranges, often down to ambient temperature, were observed (see Table S1 in the Supporting Information).

4-Substituted Compounds
Interestingly, the stability and the phase range of the Cub bi phase is enlarged by increasing the size of the substituent Scheme 1. Chemical structures of the polycatenar bithiophene derivatives 3/ m X n and 3a/Y under consideration.

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© 2020 The Authors. Published by Wiley-VCH GmbH (H < F < I), whereby electron acceptors (halogens, OCF 3 , CN, NO 2 ), reducing the electron density of the polyaromatic core, have a larger stabilizing effect than donors (CH 3 , OCH 3 ) with approximately the same volume. This becomes especially evident if 3/ 4 CN and 3/ 4 Me, both having the same volume of the substituent, but very different Hammett substituent constants (as measures for the electronic substituent effects) [129] are compared ( Table 1). Table 1 also summarizes the volumes of the different substituents as estimated using crystal volume increments (V c ) [130] and their Hammett constants (σ m , σ p ). [129] In most cases, the Cub bi phase of the 4-substituted compounds is Ia3 d, only for the OCF 3 group, the phase type changes to I23. [131] The cubic lattice parameter a cub of the Ia3 d phase of all 3/0 type tricatenar compounds 3/ 4 X is in the same range between 10.8 and 11.5 nm (Table 1 and Table S1 (Supporting Information)) and only slightly increasing with the volume of the substituent X, whereas that of the I23 phase of 3/ 4 OCF3 is about one half larger (a cub = 16.2 nm), in line with the triple network structure. Compound 3/ 4 ODec, also forming the I23 phase but belonging to the 3/1-tetracatenar compounds, will be discussed further below.

3-Substituted Compounds
Shifting the substituent X to the lateral 3-position (Figure 4a) provides reduced Cub-Iso transition temperatures compared to the 4-substituted compounds; only the polar electron acceptor substituents F, Br and especially CN can still stabilize the cubic phase compared to X = H. This cubic phase stabilizing effect of polar substituents is smaller than for the 4-substituted series, due to the competing effect of the stronger steric distortion of molecular packing by the lateral position of the substituents. The cubic phase stabilizing effect of the polar CN group is again outstanding if compared with Br, having a similar volume. The Cub bi phase type changes already for X = Br from Ia3 d to I23, indicating the stronger steric effect of the substituents in the lateral 3-position on the phase type ( Figure 4a and Table 2). Only for compound 3/ 3 OEt, the I23 phase is observed under all conditions, whereas for compounds 3/ 3 OMe, 3/ 3 Br, and 3/ 3 I, both Cub bi phase types, Ia3 d and I23, can be found depending on temperature and other conditions such as cooling rate and thermal history (for a more detailed discussion, see Section S1.4 in the Supporting Information). Overall, with increasing volume of X there is a transition from Ia3 d (X = H, F) via an I23-Ia3 d dimorphism (X = Br, I, OMe) to I23 (X = OC 2 H 5 ). In the sequence OCH 3 -OC 2 H 5 -OC 10 H 21 , i.e., by elongation of the alkyloxy chain (Figure 4a), the Ia3 d phase is at first completely removed and replaced by the I23 phase for OC 2 H 5 and then the Ia3 d phase re-emerges for OC 10 H 21 . Thus, there appear to be two Ia3 d phase ranges, one favored at small apex size and the second one requiring a significantly larger substituent at the apex; the two are separated by the I23 phase for medium effective volume of the substituent X. The transition from 3/ 3 OMe to 3/ 3 ODec (and from 3/ 4 OMe to 3/ 4 ODec, see Figure 3a) can be considered as a transition between two types of polycatenars, the taper-shaped 3/0 tricatenars and the nonsymmetric 3/1 tetracatenars. [132] As shown in Figure 5a,b, this phase sequence is also observed by POM in the contact region between the achiral Ia3 d phases of 3/H and 3/ 3 ODec where a ribbon of the chiral I23 phase is induced. Thus, tailoring of the cubic phase structure and mirror symmetry breaking is not only achievable by the design of individual compounds, but also by mixing compounds with different sizes of the substituents. [111,117]

3,5-Disubstituted Compounds
The 3,5-disubstitution pattern (Figure 4b) provides an even stronger steric effect than only one 3-substituent, and therefore in all cases, even for X = F, the Ia3 d phase of 3/H is replaced by the I23 phase. Outstanding is again the effect of the polar electron accepting NO 2 group, leading to the highest Cub bi -Iso transition temperature and the widest cubic range despite its significant size. We attribute this phase stabilization to two main effects. First, the increased dipole moment strengthens the electrostatic intermolecular interactions. Especially, the stronger core-core interaction between the electron-deficient apex and the trialkoxy-substituted electron donor ends in an antiparallel side-by-side packing of the polyaromatic rods ( Figure 3b) leads to a denser packing, stabilizing the networks. Second, the polar groups increase the incompatibility of the poly aromatic core units with the nonpolar aliphatic chains,  [87]; for all other compounds, see Table S1 in the Supporting Information; abbreviations: Cr = crystalline solid Cub/Ia3d = Cub bi phase with Ia3d lattice; all Ia3d phases represent long pitch Ia3d (L) phases; Cub/I23 = Cub bi phase with I23 lattice, M = unknown mesophase, the isotropic liquid state (Iso) is at the right side of the columns; b) space filling molecular models (CPK models) of the proposed organizations of the molecules of compound 3/ 4 F in the Cub bi phase. a) V cX = crystal volume of the substituent X, estimated with the crystal volume increments (V c ) reported by Immirzi and Perini; [130] σ m and σ p are Hammett constants (Table S11, Supporting Information); [129] σ m is a measure of the strength of electron withdrawing (+) and electron donating (−) effects via the σ bonds (inductive effect) and σ p is an analogous measure for the delocalization of the π-electrons (resonance effect); large positive values of σ m and σ p indicate electron acceptors, whereas small positive and negative values are typically found for electron donating substituents; n raft = number of molecules organized in each 0.45 nm thick raft of the networks, calculated as described in Table S10 (Supporting Information); Φ = helical twist between adjacent 0.45 nm thick rafts of molecules along the networks, calculated according to: Φ(Ia3d) = 70.5°/[0.354a cub /0.45 nm] [87] and Φ(I23) = 90°/[0.290a cub /0.45 nm]; [99] abbreviations: www.afm-journal.de www.advancedsciencenews.com

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© 2020 The Authors. Published by Wiley-VCH GmbH additionally contributing to mesophase stability by strengthened segregation. [15,16,133] In both series 3/ 3 X and 3/ 35 X 2 , there is no significant effect of the volume of the apex on the lattice parameter, it being 15.6-16.4 nm for the I23 phase and 10.4-10.9 nm for the Ia3 d phase (Table 2). Only the Ia3 d lattice of 3/ 3 ODec has a significantly smaller size (a cub = 9.8 nm) despite the fact that this compound has the largest and the longest substituent. In the fully intercalated antiparallel organization, the additional long alkyl chain is organized between the other alkyl chains. In this arrangement, the length of the pairs is not changed, but the number of alkyl chains in the periphery grows and this reduces the number of molecules in the network crosssection, which then reduces a cub .
The total length of the networks in each unit cell (L net ) can be calculated according to L net = 8.485a Ia3d for the Ia3 d phase 87 and according to L net = 20.68a I23 for the I23 phase, [99] and the number of molecules arranged along this distance is determined by L net /n cell , where n cell is the number of molecules per unit cell. Assuming a constant distance between the molecules of 0.45 nm, it is calculated according to n raft = n cell /(L net /0.45) that about four (n raft = 3.9) molecules are organized in the rafts forming the networks of 3/H and only about three (n raft = 2.7) for 3/ 3 ODec (see Table 2). This means that peripheral space filling reduces the diameter of the network segment, which additionally affects the inter-raft twist. In the Ia3 d phase, the twist between the molecules in neighboring rafts can be calculated according to Φ(Ia3 d) = 70.5°/[0.354a cub /0.45 nm] [87] and in the I23 phase as Φ(I23) = 90°/[0.290a cub /0.45 nm]. [99] Table 2 collates the lattice parameters a cub and the twist angles Φ of compounds 3/ 3 X and 3/ 35 X 2 depending on the total volume of the substituted benzene ring at the apex (V apex ). Interestingly, the calculated twist angle Φ continuously rises with expanding apex volume across the Ia3 d → I23 → Ia3 d transitions from 8.2° to 8.6° in Ia3 d via 8.5°-9.0° in I23 to 9.1° for the Ia3 d phase of 3/ 3 ODec. This means that with increasing size of the Mesophases and transition temperatures a) of compounds 3/ 3 X and b) of compounds 3/ 35 X 2 , recorded on heating; for compound 3/ 3 ODec, a Col hex phase is formed on cooling from Iso which is rapidly replaced by the Ia3d phase (see Figure S2 in the Supporting Information); the Ia3d phase of 3/ 3 ODec, indicated by a darker blue, represents a short pitch Ia3d (S) phase, whereas all others, shown in light blue belong to the long pitch Ia3d (L) type. [a] For 3/ 3 Br and 3/ 3 I only, the Ia3d phase is observed on cooling; [b] for 3/ 3 OMe on cooling, Ia3d and I23 phases were observed, depending on the condition, see Section S1.4 in the Supporting Information; the numerical data are collated in Table S1 in the Supporting Information and for 3/H and 3/ 3 OEt, see ref. [87]. Abbreviations: Ia3d (S) = short pitch Ia3d phase; for the other abbreviations, see Figure 1 and Table 1; V apex includes the benzene ring with attached substituents X (3/ m X n ) or the group Y (3a/Y); for calculations, see Table S10 in the Supporting Information.

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© 2020 The Authors. Published by Wiley-VCH GmbH apex, the helical twist becomes larger and the pitch becomes shorter and therefore a frustration between pitch length and the distance between the junctions arises in the Ia3 d phase. At a certain degree of frustration, the Ia3 d phase is replaced by the triple network structure with I23 lattice, allowing a larger twist angle Φ. However, this advantage of the I23 lattice obviously exists only for a certain range of the helical pitch and upon further increasing the volume of the apex, a mismatch arises also in I23. The I23 phase appears to be even more sensitive to any mismatch between junction distance and pitch length and after reaching this limit, the Ia3 d lattice becomes the thermodynamically more stable mode of self-assembly again. This means that there is a long pitch (Ia3 d (L) ) and a short pitch Ia3 d phase (Ia3 d (S) ) which are separated by the I23 phase.

3,4-Disubstituted Compounds
In contrast to the 3,5-disubstitution pattern (Figure 4b), 3,4-disubstitution (Figure 6a) stabilizes the cubic phases by 30-40 K compared to the 3-monosubstituted, whereas the effect on the cubic phase type is weaker than for the 3,5 disubstitution pattern. For example, the I23 phase of 3/ 3 OMe is removed and the Ia3d phase becomes dominant again for the 3,4-dimethoxy-substituted compound 3/ 34 OMe 2 . Because no induced I23 phase could be found in the contact region between the Ia3d (L) phase of compound 3/H and the Ia3d phase of 3/ 34 OMe 2 (see Figure S5 in the Supporting Information), it is confirmed to belong to the long pitch Ia3d (L) type. This indicates that 4-substitution provides a smaller twisting power and can reduce the stronger effect of the lateral substituents. This reduced twist is also evident from the comparison of the 4-and 3-decyloxy-substituted compounds 3/ 4 ODec and 3/ 3 ODec (Figures 3a and 4a) where only the 3-substituted forms the short pitch Ia3d (S) phase and the 4-substituted is still I23.  in all cases, the Ia3d phase represents a long pitch Ia3d (L) phase (for numerical data of 3/H and 3/ 34 OMe 2 , see ref. [87], for all other compounds, see Table S1 in the Supporting Information). c) Molecular structure and transition temperatures (T/°C) of compound 3/ 26 F 2 and d-f) chiral domains as observed by POM of the SmC s phase of 3/ 26 F 2 in a hometropic cell, the view is parallel to the helix axis developing between the two substrate surfaces; for textures observed by rotating the sample between the polarizers, see Figure S4 in the Supporting Information and for a planar texture, see Figure S3 in the Supporting Information.

2-Substituted and 2,6-Disubstituted Compounds
As expected, the cubic phases are destabilized by the 2-substituents at the apex, due to their especially strong distorting steric effect on the molecular packing in this position (Figure 6b). The cubic phase stability is even for the fluorinated compound 3/ 2 F and the 2-nitro-substituted compound 3/ 2 NO 2 reduced compared to 3/H, and the 2-OMe-substituted compound 3/ 2 OMe does not show any LC phase (melting point: 131 °C). The very different effect of 2-substitution by the NO 2 group compared to 3-substitution becomes obvious by comparing Figures 4b  and 6b. It appears that in 2-position, the mesophase stabilizing effect of this electron acceptor group is switched off and only the destabilizing steric effect remains. This might be due to a twist introduced around the COO group which hinders the electrostatic interaction with the sterically demanding trialkoxysubstituted donor ring in the antiparallel packing. That even in this case the Cub bi phase is retained demonstrates clearly the robustness of the design concept of 3/0 tricatenars. While the 2-substitution strongly reduces the stability of the cubic phase, there is no change in the cubic phase type; even 3/ 2 I forms exclusively the long pitch Ia3 d (L) phase, as confirmed by the induction of a chiral I23 phase in the contact region with the Ia3 d (S) phase of 3a/CEt 3 (Figure 5c,d). By contrast, for the isomeric laterally 3-substituted compound 3/ 3 I, the I23 phase dominates (Figure 4a).
Compound 3/ 26 F 2 having two ortho-fluorines in the 2-and 6-positions does not form any cubic phase. This compound melts at 118 °C and shows only a monotropic and synclinic tilted SmC s phase on cooling below 112 °C (Figure 6c). A special feature of the SmC s phase of 3/ 26 F 2 is the formation of chiral domains in thin homeotropically aligning cells, indicated by an optical texture showing a conglomerate of dark and bright domains inverting their brightness upon inverting the twist sense of the analyzer (Figure 6d-f) and remaining unchanged by rotation of the sample between crossed polarizers ( Figure S4, Supporting Information). [134,135] In the case of compound 3/ 26 F 2 , the steric effects of the two ortho-fluorines can obviously stabilize helical conformers around the COO group, [136] thus supporting chirality synchronization by helix formation between the substrate surfaces. [137] Overall, substitution in the 2-position has a strong destabilizing effect on the Cub bi phase without significantly affecting the cubic space group.

3,4,5-Trisubstituted and 2,3,4,5,6-Pentasubstituted Compounds
The effect of 3,4,5-trisubstitution is shown in Figure 7 for F and OMe substituents as examples. Only the I23 type Cub bi phase is observed for both compounds. Again, the additional fluorine in the 4-position of the trisubstituted compound 3/ 345 F 3 enhances the cubic phase stability compared to the 3,5-disubstituted compound 3/ 35 F 2 by 25 K (Figure 4b). A similar stabilization is observed by comparing the methoxy-substituted compounds 3/ 35 OMe 2 (Figure 4b) and 3/ 345 OMe 3 (Figure 7). Even the pentafluorobenzoate 3/ 23456 F 5 shows exclusively a broad Cub bi phase over a range exceeding that of the nonfluorinated compound 3/H with a smaller apex. This is mainly attributed to the electron withdrawing effect of the fluorines, increasing the polar core-core interactions. That the steric effect of fluorination is moderate is indicated by the fact that only the I23 phase, but not the short pitch Ia3 d (S) phase is observed for the polyfluorinated compounds. Thus, the taper-shaped tricatenars tolerate multiple substitutions at the apex, allowing easy access to π-systems with distinct electronic properties, while retaining their organization in a well-defined 3D network with cubic symmetry. The major general trends provided by the different apex structures and their substitution patterns are summarized in Figure 8.

Heterocycles and Alicycles at the Apex
Whereas no effect of replacement of benzene at the apex by thiophene is observed, replacing benzene by cyclohexane changes the cubic phase type from Ia3 d to I23, indicating that the aliphatic ring acts as a larger substituent, especially  [87], for compounds 3a/Cy, 3a/Ad and 3a/CEt 3 , see ref. [117], for all other data, see Tables S1 and S2 in the Supporting Information).  Table 2). It is remarkable that even for the large polycyclic adamantyl group (3a/Ad), the I23 phase is retained with almost the same Cub bi -Iso transition temperature as for the cyclohexyl-substituted compound 3a/Cy. [117] Even the triethylacetate 3a/CEt 3 , which can be considered as a noncyclic ring-opened relative of the adamantane compound 3a/Ad, forms a cubic phase, though with reduced stability. [117] That these large substituents at the apex cannot remove the cubic phase confirms once more the robustness of this molecular design concept. In contrast to 3a/Ad with an I23 phase, the cubic phase of 3a/CEt 3 has an Ia3 d space group. The small cubic lattice parameter of 3a/CEt 3 (a cub = 8.85 nm) and the large twist (Φ = 10.1°) are in line with a short pitch Ia3 d (S) phase, which is confirmed by the induction of a chiral I23 phase in the contact region with the Ia3 d (L) phase of 3/H (see Figure S6 in the Supporting Information). It shows that the triethylacetate group of 3a/CEt 3 acts as an even larger substituent than adamantane. The larger effective size should be the result of the increased conformational flexibility compared to adamantane, allowing more bulky conformations to be involved in the equilibrium. The change of the cubic phase type upon increasing the cycloaliphatic ring size from cyclobutyl to cyclododecyl is described in more detail in a previous communication. [117] Overall, these bulky aliphatic apex structures have a strong steric effect on mesophase stability, as well as a strong effect on network helicity.

Condensed Aromatics and Biphenyl at the Apex
Broad regions of the Ia3 d (L) phase were found for the two naphthalene derivatives, where the higher transition temperatures were observed for the 2-naphthoic acid derivative with the naphthyl group arranged colinear with the rod-like core (Figure 7). By contrast, the side-on attached 1-substituted naphthalene leads to a reduction of the cubic phase stability and the anthracene-9-carboxylate 3a/Anth with two laterally annulated benzene rings does not show any LC phase (Figure 7). Based on the calculated twist angles Φ ( Table 2) and in line with the results of the optical investigation of the contact regions with the Ia3 d (L) and Ia3 d (S) phases of related compounds (see Figures S7 and S8 and the additional discussions in Section S1.4.3 in the Supporting Information), the cubic phase of compounds 3a/1-Npht, 3a/2-Npht, and 3a/Biph is of the same long pitch Ia3 d (L) type. Elongation of the rod-like core by the coaxial 2-napthyl unit, or by replacing the single benzene ring by a linear biphenyl unit stabilizes the cubic phase (Figure 7), increases the lattice parameter, and reduces the twist angle between adjacent molecules along the networks segment if compared to 3/H (Table 2). Overall, there is a very different behavior of aromatic and cycloaliphatic [117] end groups. The flat and rigid aromatics contribute to the length and volume of the core unit and thus reduce the twist angle between the cores, retaining the Ia3 d (L) phase. By contrast, aliphatic and cycloaliphatic units tend to segregate from the core and contribute to the space filling in the aliphatic periphery, thus increasing the twist, disfavoring the Ia3 d (L) and tending toward I23 and ultimately to Ia3 d (S) .

Helical Twist and Cubic Phase Formation
The lateral distance between the two networks in the Ia3d phase, corresponding to the distance between the minimal surfaces, and calculated as d net = √3a cub /4 = 4.7-4.8 nm, is only a little bit larger than L mol (4.4-4.6 nm for most compounds). The antiparallel organization of the molecules with full intercalation of the rod-like cores (Figure 3b) would lead to a length of these pairs of around 6.1 nm. The difference to d net is considered to be due to partial alkyl chain intercalation and chain folding to efficiently fill the space around the networks. The interfacial curvature provided by the 2:3 ratio of core to alkyl chain cross-sectional areas at the interfaces in this organization of the 3/0 tricatenars (Figure 9b,c) appears to be just right for the formation of Cub bi phases. A larger 2:4 ratio is obtained for the tetracatenars (Figure 9d,e). Though the 3/1 tetracatenars (Figure 9d) form predominately Cub bi phases too, the symmetric 2/2 tetracatenars (Figure 9e) are much more sensitive to structural modifications and have, if any, only small Cub bi ranges beside the dominant lamellar and columnar phases. [69,116,[120][121][122][123][124] One may argue that, on average, rafts of antiparallel 3/1 tetracatenars are equivalent to those of 2/2 Figure 8. Summary of the relations between molecular structure, helical twist, cubic phase type, and cubic phase stability of compounds 3/ m X n and 3a/Y; the slight overlap of the helical twist angles results from uncertainties in the determination of the molecular volumes by the used increments [130] and possible effects of electrostatic core-core interactions, additionally contributing to the self-assembly.

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© 2020 The Authors. Published by Wiley-VCH GmbH tetracatenars. However, as the ribbon winds it's way through the surrounding of other twisted ribbons, its immediate environment changes and while at one point a symmetric arrangement may be ideal, at other points, more chains may be needed on one side of the ribbon than on the other to plug a local void. Symmetric polycatenars cannot adjust to such local steric requirements. However, it is surprising that such a huge variety of different types of apex substitution, even involving extremely bulky groups as adamantane (3a/Ad), pentafluorobenzene (3/ 23456 F 5 ), naphthalene (3a/1-Npht), and highly branched alkyl chains (3a/CEt 3 ) can lead to such broad temperature ranges of Cub bi network phases (Figure 7). It appears that these bulky end groups, favoring the helical twist, allow a denser molecular packing due to an at least partial synchronization of the overall helical molecular conformers. [86,95,96,138] This seems to provide an additional advantage compared to competing lamellar phases, disfavoring the twist, and the columnar phases, having only short range helix correlation. [95,96,139,140] If the enthalpic gain of uniform helix packing, provided by the long range transmission of helicity by network formation, exceeds the entropic penalty of chirality synchronization, then the Cub bi networks can be additionally stabilized over the competing lamellar and columnar modes of self-assembly. It appears that this Cub bi phase stabilizing effect of polyaromatic cores is more general. For example, robust bicontinuous cubic phases were even found for polyaromatic rods with only two alkyl chains (Figure 9a, e.g., BABHs [107,108] and the hydrogen-bonded dimers of the ANBCs, see Figure 1c,d). [16,101,103,104] In these cases, alkyl chain intercalation or replacing the alkyl chains by perfluorinated chains [141,142] or by adding bulky silyl groups [74,108] provides the required interfacial curvature for Cub bi phase formation. Permanent molecular chirality can further support the helical twist, leading to a series of chirality frustrated LC phases. [53][54][55]143] This contribution of helical packing to Cub bi phase stabilization is absent for flexible amphiphilic molecules [7] and their dendritic [18,144] and polymeric analogs (block copolymers) [10] which limits their capability of Cub bi phase formation to narrower ranges and this explains the unexpected Cub bi phase stability of properly designed rod-like compounds.
Polar substituents like CN and NO 2 with the highest polarity and electron acceptor properties (large σ m and σ p Hammett constants, see Table 1 and Table S11 in the Supporting Information) show an especially strong stabilizing effect on the cubic phases due to dipolar interactions and the strengthening of the electrostatic interactions between the electron-rich trialkoxylated and the electron-poor CN-/NO 2 -substituted ends. It appears that polar 3-and 4-substitutions favor a dense packing and increase the intermolecular twist, whereas 2-substitution predominately favors intramolecular twist and reduces the core packing density for steric reasons, thus favoring a transition to lamellar phases ( Figure 8). However, only the double ortho-substitution of the apex in 3/ 26 F 2 and the lateral annulation by two benzene rings in compound 3a/Anth can remove the cubic phases completely.
Further increasing the polarity of the apex substituents, as for example, by introduction of ionic groups or hydrogen bonding groups has the opposite effect and leads, as shown previously, to hexagonal columnar and micellar cubic phases (Cub sph ), even for compounds with a long rigid core (Figure 9g). [145][146][147][148][149][150][151][152] In all these cases, the segregation of the polar apex from the aromatic cores sets in, which inhibits the antiparallel side-byside packing of the aromatic cores (Figure 9f). This changes the ratio of aromatic to aliphatic cross-section areas from 2:3 to 1:3, leading to a much stronger interfacial curvature, favoring columnar and micellar self-assembly. Similarly, reducing the rod length and thus increasing the taper angle supports higher interfacial curvature, too. [18][19][20][21]75,144,145,153]

Helical Twist and the Ia3 d (L) -I23-Ia3 d (S) -Cubic Phase Sequence
For the tapered 3/0 tricatenars reported herein, there is a strong dependence of the space group on the size and position of the  (Figure 8). [132] In line with the double network structure of the Ia3 d phase and a triple network structure of the I23 phase, the lattice parameter increases by about 50 ± 10% at the Ia3 d (L) -to-I23 transitions. However, within the Ia3 d and the I23 phases, the cubic lattice parameter is almost independent of the size of the apex substituent. There is obviously a compensation of the increasing molecular volume, tending to increase a cub , opposed by the increasing twist angle between successive rafts and decreasing the number of molecules in a raft, both contributing to a reduction in a cub (Tables 1 and 2 and Table S10 in the Supporting Information). Because the molecules in the twisted ribbons must arrive almost perpendicular to the plane of the network junction, a frustration arises between optimal helical pitch and the junction distance. At the same time, the twist angle Φ between successive rafts is a compromise between Φ-increasing tendency of the clashing end groups and the Φ-reducing tendency of the rod-like cores, which prefer to lie parallel and maximize their π-π interaction. Another contribution comes from the diastereomeric relation and coupling between supramolecular helical twist and transient molecular helicity. All these affect the choice of the cubic phase type. [87,95] Each structure in the sequence Ia3 d (L) -I23-Ia3 d (S) represents the solution best suited to accommodate a certain range of the twist angle Φ. With increasing apex size, the increasing Φ reaches up to about 8.7° for the long pitch Ia3 d (L) phase, 8.5°-9.2° for the I23 phase, and ≥9.1° for the short pitch Ia3 d (S) phase (see Figure 8, Tables 1 and 2, and Table S10 in the Supporting Information). These values refer to molecules involving the bisbenzoylated 5,5′-diphenyl-2,2′-bithiophene core unit. They can deviate from these values, for example, if the alkyl chain at the apex is especially long (X = OC 10 H 21 ), or as the length and shape of the core is changed, though the fundamental phase sequence remains the same. [117] Depending on the contribution of the substituent to the volume effect and the twist, the lattice parameter can increase, as observed for the BABHs and ANBCs, [101,103,104,107,108] remain almost constant, as reported here for compounds 3/ m X n and 3a/Y, or even decreases with increasing volume of the substituents, as recently reported for benzil-based polycatenars. [154] However, in all cases, the twist angle increases with the apex volume, leading to the sequence Ia3 d (L) -I23-Ia3 d (S) . While the cubic phase destabilizing steric effect of apex substitution increases from the 4-via the 3-to the 2-position, the effect on the Cub bi phase structure is the largest for the sidewise-directed lateral 3-position and is smaller for the inward-directed lateral 2-and the terminal 4-position ( Figure 8).

Summary and Conclusions
Overall, the concept of taper-shaped 3/0 tricatenars, involving an extended rod-like unit, represents a new robust platform for the design of LC network structures with cubic symmetry. In most cases, the Cub bi phases (Ia3 d and I23), representing interwoven networks with 3-way junctions, are the only observed LC phases in wide temperature ranges and for a wide variety of groups attached to the apex, including perfluorinated aromatics, heteroaromatics, alicyclic and polycyclic rings, and even for highly branched chains (Figures 3, 4, 6-8). Often, these network phases are stable down to ambient temperature (Tables S1 and S2, Supporting Information). An organization with locally adjustable antiparallel side-by-side packing of the rod-like cores, leading to a 2:3 average ratio of core-to-chain width (Figures 3b and 9b,c), is proposed to be responsible for the robustness of the Cub bi phases and their strongly preferred formation over lamellar and columnar modes of organization. The chirality-synchronized helical packing in the networks is thought to additionally stabilize the cubic phases by allowing a denser molecular packing. The observed insensitivity of cubic network formation to apex substitution by a large diversity of functional groups allows tuning of the electronic properties (HOMO-LUMO bandgap) of the π-conjugated rod-like cores organized in the helical networks. This potential for application as functional materials is augmented by the possibility of switching between an achiral (Ia3 d) and chiral (I23) state. With increasing size of the apex the sequence Ia3 d (L) -I23-Ia3 d (S) is observed which appears to be universal and is explained by an increasing helical twist along the networks (see Figure 8, Tables 1 and 2, and Table S10 in the Supporting Information). The long pitch and short pitch Ia3 d phases are considered as distinct ranges of the same macroscopically achiral gyroid double network phase, which is interrupted for a certain twist angle range by the homochiral triple network I23 phase (Figure 8). The helical twist angle can be adjusted by design of the individual molecules or, alternatively, by mixing short and long pitch Ia3 d phases ( Figure 5).
Overall, this work provides a general understanding of supramolecular self-assembly in Cub bi network phases depending on details of the molecular structure and thus paves the way to the directed design of cubogenic and spontaneously chiral functional materials with tunable application-relevant properties. More generally, it provides insights into the process of spontaneous mirror symmetry breaking in soft matter, [95,96,136,155] and it shows the importance of network formation for this process. Spontaneous emergence of chirality during self-assembly is also of paramount interest for circularly-polarized emission [46][47][48] and photonic applications, [27,28] and for new dynamic routes to absolute enantioselective synthesis. [156][157][158]

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.